Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.7s

Alternatives: 10

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + y) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (fma (log y) (- -0.5 y) (- y z))))

C

double code(double x, double y, double z) {
	return x + fma(log(y), (-0.5 - y), (y - z));
}

Julia

function code(x, y, z)
	return Float64(x + fma(log(y), Float64(-0.5 - y), Float64(y - z)))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

   2. sub-neg 99.8%

      \[\leadsto \color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + \left(y - z\right) \]

   3. associate-+l+ 99.9%

      \[\leadsto \color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)\right)} \]

   4. *-commutative 99.9%

      \[\leadsto x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + \left(y - z\right)\right) \]

   5. distribute-rgt-neg-in 99.9%

      \[\leadsto x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + \left(y - z\right)\right) \]

   6. fma-def 99.9%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]

   7. +-commutative 99.9%

      \[\leadsto x + \mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y - z\right) \]

   8. distribute-neg-in 99.9%

      \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y - z\right) \]

   9. unsub-neg 99.9%

      \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y - z\right) \]

   10. metadata-eval 99.9%

       \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]

4. Final simplification 99.9%

   \[\leadsto x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right) \]

Alternative 2: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ \mathbf{if}\;y \leq 2.85 \cdot 10^{-263}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+41}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+170}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (log y) -0.5) z)))
   (if (<= y 2.85e-263)
     (- x z)
     (if (<= y 4.1e-190)
       t_0
       (if (<= y 7.7e-47)
         (+ x (- y z))
         (if (<= y 4.9e-31)
           t_0
           (if (<= y 1.15e+41)
             (- x z)
             (if (<= y 1.15e+170)
               (+ x (* y (- 1.0 (log y))))
               (- y (+ z (* y (log y))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * -0.5) - z;
	double tmp;
	if (y <= 2.85e-263) {
		tmp = x - z;
	} else if (y <= 4.1e-190) {
		tmp = t_0;
	} else if (y <= 7.7e-47) {
		tmp = x + (y - z);
	} else if (y <= 4.9e-31) {
		tmp = t_0;
	} else if (y <= 1.15e+41) {
		tmp = x - z;
	} else if (y <= 1.15e+170) {
		tmp = x + (y * (1.0 - log(y)));
	} else {
		tmp = y - (z + (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (log(y) * (-0.5d0)) - z
    if (y <= 2.85d-263) then
        tmp = x - z
    else if (y <= 4.1d-190) then
        tmp = t_0
    else if (y <= 7.7d-47) then
        tmp = x + (y - z)
    else if (y <= 4.9d-31) then
        tmp = t_0
    else if (y <= 1.15d+41) then
        tmp = x - z
    else if (y <= 1.15d+170) then
        tmp = x + (y * (1.0d0 - log(y)))
    else
        tmp = y - (z + (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.log(y) * -0.5) - z;
	double tmp;
	if (y <= 2.85e-263) {
		tmp = x - z;
	} else if (y <= 4.1e-190) {
		tmp = t_0;
	} else if (y <= 7.7e-47) {
		tmp = x + (y - z);
	} else if (y <= 4.9e-31) {
		tmp = t_0;
	} else if (y <= 1.15e+41) {
		tmp = x - z;
	} else if (y <= 1.15e+170) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else {
		tmp = y - (z + (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(y) * -0.5) - z
	tmp = 0
	if y <= 2.85e-263:
		tmp = x - z
	elif y <= 4.1e-190:
		tmp = t_0
	elif y <= 7.7e-47:
		tmp = x + (y - z)
	elif y <= 4.9e-31:
		tmp = t_0
	elif y <= 1.15e+41:
		tmp = x - z
	elif y <= 1.15e+170:
		tmp = x + (y * (1.0 - math.log(y)))
	else:
		tmp = y - (z + (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * -0.5) - z)
	tmp = 0.0
	if (y <= 2.85e-263)
		tmp = Float64(x - z);
	elseif (y <= 4.1e-190)
		tmp = t_0;
	elseif (y <= 7.7e-47)
		tmp = Float64(x + Float64(y - z));
	elseif (y <= 4.9e-31)
		tmp = t_0;
	elseif (y <= 1.15e+41)
		tmp = Float64(x - z);
	elseif (y <= 1.15e+170)
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	else
		tmp = Float64(y - Float64(z + Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(y) * -0.5) - z;
	tmp = 0.0;
	if (y <= 2.85e-263)
		tmp = x - z;
	elseif (y <= 4.1e-190)
		tmp = t_0;
	elseif (y <= 7.7e-47)
		tmp = x + (y - z);
	elseif (y <= 4.9e-31)
		tmp = t_0;
	elseif (y <= 1.15e+41)
		tmp = x - z;
	elseif (y <= 1.15e+170)
		tmp = x + (y * (1.0 - log(y)));
	else
		tmp = y - (z + (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 2.85e-263], N[(x - z), $MachinePrecision], If[LessEqual[y, 4.1e-190], t$95$0, If[LessEqual[y, 7.7e-47], N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e-31], t$95$0, If[LessEqual[y, 1.15e+41], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.15e+170], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(z + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 2.8499999999999999e-263 or 4.90000000000000023e-31 < y < 1.1499999999999999e41

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.9%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in z around inf 78.9%

      \[\leadsto x - \color{blue}{z} \]

   if 2.8499999999999999e-263 < y < 4.1000000000000002e-190 or 7.6999999999999999e-47 < y < 4.90000000000000023e-31

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + \left(y - z\right) \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)\right)} \]

      4. *-commutative 100.0%

         \[\leadsto x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + \left(y - z\right)\right) \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + \left(y - z\right)\right) \]

      6. fma-def 100.0%

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]

      7. +-commutative 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y - z\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y - z\right) \]

      9. unsub-neg 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y - z\right) \]

      10. metadata-eval 100.0%

          \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{\left(x + -0.5 \cdot \log y\right) - z} \]

   5. Taylor expanded in x around 0 89.8%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   6. Step-by-step derivation

      1. *-commutative 89.8%

         \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   7. Simplified 89.8%

      \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   if 4.1000000000000002e-190 < y < 7.6999999999999999e-47

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 100.0%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 99.7%

         \[\leadsto x - \left(\color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}} - \left(y - z\right)\right) \]

      2. pow3 99.7%

         \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}} - \left(y - z\right)\right) \]

      3. *-commutative 99.7%

         \[\leadsto x - \left({\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3} - \left(y - z\right)\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}} - \left(y - z\right)\right) \]

   6. Taylor expanded in y around inf 78.7%

      \[\leadsto x - \color{blue}{\left(z + -1 \cdot y\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 78.7%

         \[\leadsto x - \left(z + \color{blue}{\left(-y\right)}\right) \]

      2. sub-neg 78.7%

         \[\leadsto x - \color{blue}{\left(z - y\right)} \]

   8. Simplified 78.7%

      \[\leadsto x - \color{blue}{\left(z - y\right)} \]

   if 1.1499999999999999e41 < y < 1.15e170

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.7%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.7%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in y around inf 84.9%

      \[\leadsto x - \color{blue}{y \cdot \left(-1 \cdot \log \left(\frac{1}{y}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. sub-neg 84.9%

         \[\leadsto x - y \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{y}\right) + \left(-1\right)\right)} \]

      2. mul-1-neg 84.9%

         \[\leadsto x - y \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} + \left(-1\right)\right) \]

      3. log-rec 84.9%

         \[\leadsto x - y \cdot \left(\left(-\color{blue}{\left(-\log y\right)}\right) + \left(-1\right)\right) \]

      4. remove-double-neg 84.9%

         \[\leadsto x - y \cdot \left(\color{blue}{\log y} + \left(-1\right)\right) \]

      5. metadata-eval 84.9%

         \[\leadsto x - y \cdot \left(\log y + \color{blue}{-1}\right) \]

   6. Simplified 84.9%

      \[\leadsto x - \color{blue}{y \cdot \left(\log y + -1\right)} \]

   if 1.15e170 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.6%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.6%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in x around 0 91.4%

      \[\leadsto \color{blue}{y - \left(z + \log y \cdot \left(0.5 + y\right)\right)} \]

   5. Taylor expanded in y around inf 91.4%

      \[\leadsto y - \left(z + \color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 91.4%

         \[\leadsto y - \left(z + \color{blue}{\left(-y \cdot \log \left(\frac{1}{y}\right)\right)}\right) \]

      2. distribute-rgt-neg-in 91.4%

         \[\leadsto y - \left(z + \color{blue}{y \cdot \left(-\log \left(\frac{1}{y}\right)\right)}\right) \]

      3. log-rec 91.4%

         \[\leadsto y - \left(z + y \cdot \left(-\color{blue}{\left(-\log y\right)}\right)\right) \]

      4. remove-double-neg 91.4%

         \[\leadsto y - \left(z + y \cdot \color{blue}{\log y}\right) \]

   7. Simplified 91.4%

      \[\leadsto y - \left(z + \color{blue}{y \cdot \log y}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{-263}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-190}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-31}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+41}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+170}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \]

Alternative 3: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ \mathbf{if}\;y \leq 1.1 \cdot 10^{-263}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-47}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+41}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (log y) -0.5) z)))
   (if (<= y 1.1e-263)
     (- x z)
     (if (<= y 3.1e-190)
       t_0
       (if (<= y 4.3e-47)
         (+ x (- y z))
         (if (<= y 5.4e-31)
           t_0
           (if (<= y 2.3e+41) (- x z) (+ x (* y (- 1.0 (log y)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * -0.5) - z;
	double tmp;
	if (y <= 1.1e-263) {
		tmp = x - z;
	} else if (y <= 3.1e-190) {
		tmp = t_0;
	} else if (y <= 4.3e-47) {
		tmp = x + (y - z);
	} else if (y <= 5.4e-31) {
		tmp = t_0;
	} else if (y <= 2.3e+41) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (log(y) * (-0.5d0)) - z
    if (y <= 1.1d-263) then
        tmp = x - z
    else if (y <= 3.1d-190) then
        tmp = t_0
    else if (y <= 4.3d-47) then
        tmp = x + (y - z)
    else if (y <= 5.4d-31) then
        tmp = t_0
    else if (y <= 2.3d+41) then
        tmp = x - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.log(y) * -0.5) - z;
	double tmp;
	if (y <= 1.1e-263) {
		tmp = x - z;
	} else if (y <= 3.1e-190) {
		tmp = t_0;
	} else if (y <= 4.3e-47) {
		tmp = x + (y - z);
	} else if (y <= 5.4e-31) {
		tmp = t_0;
	} else if (y <= 2.3e+41) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(y) * -0.5) - z
	tmp = 0
	if y <= 1.1e-263:
		tmp = x - z
	elif y <= 3.1e-190:
		tmp = t_0
	elif y <= 4.3e-47:
		tmp = x + (y - z)
	elif y <= 5.4e-31:
		tmp = t_0
	elif y <= 2.3e+41:
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * -0.5) - z)
	tmp = 0.0
	if (y <= 1.1e-263)
		tmp = Float64(x - z);
	elseif (y <= 3.1e-190)
		tmp = t_0;
	elseif (y <= 4.3e-47)
		tmp = Float64(x + Float64(y - z));
	elseif (y <= 5.4e-31)
		tmp = t_0;
	elseif (y <= 2.3e+41)
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(y) * -0.5) - z;
	tmp = 0.0;
	if (y <= 1.1e-263)
		tmp = x - z;
	elseif (y <= 3.1e-190)
		tmp = t_0;
	elseif (y <= 4.3e-47)
		tmp = x + (y - z);
	elseif (y <= 5.4e-31)
		tmp = t_0;
	elseif (y <= 2.3e+41)
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 1.1e-263], N[(x - z), $MachinePrecision], If[LessEqual[y, 3.1e-190], t$95$0, If[LessEqual[y, 4.3e-47], N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-31], t$95$0, If[LessEqual[y, 2.3e+41], N[(x - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.1e-263 or 5.40000000000000027e-31 < y < 2.2999999999999998e41

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.9%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in z around inf 78.9%

      \[\leadsto x - \color{blue}{z} \]

   if 1.1e-263 < y < 3.09999999999999993e-190 or 4.2999999999999998e-47 < y < 5.40000000000000027e-31

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + \left(y - z\right) \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)\right)} \]

      4. *-commutative 100.0%

         \[\leadsto x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + \left(y - z\right)\right) \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + \left(y - z\right)\right) \]

      6. fma-def 100.0%

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]

      7. +-commutative 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y - z\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y - z\right) \]

      9. unsub-neg 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y - z\right) \]

      10. metadata-eval 100.0%

          \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{\left(x + -0.5 \cdot \log y\right) - z} \]

   5. Taylor expanded in x around 0 89.8%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   6. Step-by-step derivation

      1. *-commutative 89.8%

         \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   7. Simplified 89.8%

      \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   if 3.09999999999999993e-190 < y < 4.2999999999999998e-47

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 100.0%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 99.7%

         \[\leadsto x - \left(\color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}} - \left(y - z\right)\right) \]

      2. pow3 99.7%

         \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}} - \left(y - z\right)\right) \]

      3. *-commutative 99.7%

         \[\leadsto x - \left({\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3} - \left(y - z\right)\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}} - \left(y - z\right)\right) \]

   6. Taylor expanded in y around inf 78.7%

      \[\leadsto x - \color{blue}{\left(z + -1 \cdot y\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 78.7%

         \[\leadsto x - \left(z + \color{blue}{\left(-y\right)}\right) \]

      2. sub-neg 78.7%

         \[\leadsto x - \color{blue}{\left(z - y\right)} \]

   8. Simplified 78.7%

      \[\leadsto x - \color{blue}{\left(z - y\right)} \]

   if 2.2999999999999998e41 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.7%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.7%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in y around inf 82.5%

      \[\leadsto x - \color{blue}{y \cdot \left(-1 \cdot \log \left(\frac{1}{y}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. sub-neg 82.5%

         \[\leadsto x - y \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{y}\right) + \left(-1\right)\right)} \]

      2. mul-1-neg 82.5%

         \[\leadsto x - y \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} + \left(-1\right)\right) \]

      3. log-rec 82.5%

         \[\leadsto x - y \cdot \left(\left(-\color{blue}{\left(-\log y\right)}\right) + \left(-1\right)\right) \]

      4. remove-double-neg 82.5%

         \[\leadsto x - y \cdot \left(\color{blue}{\log y} + \left(-1\right)\right) \]

      5. metadata-eval 82.5%

         \[\leadsto x - y \cdot \left(\log y + \color{blue}{-1}\right) \]

   6. Simplified 82.5%

      \[\leadsto x - \color{blue}{y \cdot \left(\log y + -1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-263}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-190}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-47}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-31}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+41}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 4: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+48}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+148} \lor \neg \left(y \leq 1.15 \cdot 10^{+169}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.5e+48)
   (+ x (- y z))
   (if (or (<= y 7e+148) (not (<= y 1.15e+169)))
     (* y (- 1.0 (log y)))
     (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e+48) {
		tmp = x + (y - z);
	} else if ((y <= 7e+148) || !(y <= 1.15e+169)) {
		tmp = y * (1.0 - log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.5d+48) then
        tmp = x + (y - z)
    else if ((y <= 7d+148) .or. (.not. (y <= 1.15d+169))) then
        tmp = y * (1.0d0 - log(y))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.5e+48) {
		tmp = x + (y - z);
	} else if ((y <= 7e+148) || !(y <= 1.15e+169)) {
		tmp = y * (1.0 - Math.log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.5e+48:
		tmp = x + (y - z)
	elif (y <= 7e+148) or not (y <= 1.15e+169):
		tmp = y * (1.0 - math.log(y))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.5e+48)
		tmp = Float64(x + Float64(y - z));
	elseif ((y <= 7e+148) || !(y <= 1.15e+169))
		tmp = Float64(y * Float64(1.0 - log(y)));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.5e+48)
		tmp = x + (y - z);
	elseif ((y <= 7e+148) || ~((y <= 1.15e+169)))
		tmp = y * (1.0 - log(y));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.5e+48], N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 7e+148], N[Not[LessEqual[y, 1.15e+169]], $MachinePrecision]], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.49999999999999987e48

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 100.0%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 99.5%

         \[\leadsto x - \left(\color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}} - \left(y - z\right)\right) \]

      2. pow3 99.5%

         \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}} - \left(y - z\right)\right) \]

      3. *-commutative 99.5%

         \[\leadsto x - \left({\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3} - \left(y - z\right)\right) \]

   5. Applied egg-rr 99.5%

      \[\leadsto x - \left(\color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}} - \left(y - z\right)\right) \]

   6. Taylor expanded in y around inf 73.5%

      \[\leadsto x - \color{blue}{\left(z + -1 \cdot y\right)} \]
   7. Step-by-step derivation

      1. neg-mul-1 73.5%

         \[\leadsto x - \left(z + \color{blue}{\left(-y\right)}\right) \]

      2. sub-neg 73.5%

         \[\leadsto x - \color{blue}{\left(z - y\right)} \]

   8. Simplified 73.5%

      \[\leadsto x - \color{blue}{\left(z - y\right)} \]

   if 2.49999999999999987e48 < y < 6.9999999999999998e148 or 1.15e169 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.6%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.6%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in x around 0 82.0%

      \[\leadsto \color{blue}{y - \left(z + \log y \cdot \left(0.5 + y\right)\right)} \]

   5. Taylor expanded in y around inf 65.1%

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 65.1%

         \[\leadsto y \cdot \left(1 - \color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)}\right) \]

      2. log-rec 65.1%

         \[\leadsto y \cdot \left(1 - \left(-\color{blue}{\left(-\log y\right)}\right)\right) \]

      3. remove-double-neg 65.1%

         \[\leadsto y \cdot \left(1 - \color{blue}{\log y}\right) \]

   7. Simplified 65.1%

      \[\leadsto \color{blue}{y \cdot \left(1 - \log y\right)} \]

   if 6.9999999999999998e148 < y < 1.15e169

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 100.0%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in z around inf 85.4%

      \[\leadsto x - \color{blue}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{+48}:\\ \;\;\;\;x + \left(y - z\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+148} \lor \neg \left(y \leq 1.15 \cdot 10^{+169}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 5: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+41}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+169}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.95e+41)
   (- (+ x (* (log y) -0.5)) z)
   (if (<= y 9e+169) (+ x (* y (- 1.0 (log y)))) (- y (+ z (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e+41) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if (y <= 9e+169) {
		tmp = x + (y * (1.0 - log(y)));
	} else {
		tmp = y - (z + (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.95d+41) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if (y <= 9d+169) then
        tmp = x + (y * (1.0d0 - log(y)))
    else
        tmp = y - (z + (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e+41) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if (y <= 9e+169) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else {
		tmp = y - (z + (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.95e+41:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif y <= 9e+169:
		tmp = x + (y * (1.0 - math.log(y)))
	else:
		tmp = y - (z + (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.95e+41)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif (y <= 9e+169)
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	else
		tmp = Float64(y - Float64(z + Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.95e+41)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif (y <= 9e+169)
		tmp = x + (y * (1.0 - log(y)));
	else
		tmp = y - (z + (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.95e+41], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 9e+169], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(z + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9499999999999998e41

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + \left(y - z\right) \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)\right)} \]

      4. *-commutative 100.0%

         \[\leadsto x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + \left(y - z\right)\right) \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + \left(y - z\right)\right) \]

      6. fma-def 100.0%

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]

      7. +-commutative 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y - z\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y - z\right) \]

      9. unsub-neg 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y - z\right) \]

      10. metadata-eval 100.0%

          \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]

   4. Taylor expanded in y around 0 97.7%

      \[\leadsto \color{blue}{\left(x + -0.5 \cdot \log y\right) - z} \]

   if 1.9499999999999998e41 < y < 8.9999999999999999e169

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.7%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.7%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in y around inf 84.9%

      \[\leadsto x - \color{blue}{y \cdot \left(-1 \cdot \log \left(\frac{1}{y}\right) - 1\right)} \]
   5. Step-by-step derivation

      1. sub-neg 84.9%

         \[\leadsto x - y \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{y}\right) + \left(-1\right)\right)} \]

      2. mul-1-neg 84.9%

         \[\leadsto x - y \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} + \left(-1\right)\right) \]

      3. log-rec 84.9%

         \[\leadsto x - y \cdot \left(\left(-\color{blue}{\left(-\log y\right)}\right) + \left(-1\right)\right) \]

      4. remove-double-neg 84.9%

         \[\leadsto x - y \cdot \left(\color{blue}{\log y} + \left(-1\right)\right) \]

      5. metadata-eval 84.9%

         \[\leadsto x - y \cdot \left(\log y + \color{blue}{-1}\right) \]

   6. Simplified 84.9%

      \[\leadsto x - \color{blue}{y \cdot \left(\log y + -1\right)} \]

   if 8.9999999999999999e169 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.6%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.6%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in x around 0 91.4%

      \[\leadsto \color{blue}{y - \left(z + \log y \cdot \left(0.5 + y\right)\right)} \]

   5. Taylor expanded in y around inf 91.4%

      \[\leadsto y - \left(z + \color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 91.4%

         \[\leadsto y - \left(z + \color{blue}{\left(-y \cdot \log \left(\frac{1}{y}\right)\right)}\right) \]

      2. distribute-rgt-neg-in 91.4%

         \[\leadsto y - \left(z + \color{blue}{y \cdot \left(-\log \left(\frac{1}{y}\right)\right)}\right) \]

      3. log-rec 91.4%

         \[\leadsto y - \left(z + y \cdot \left(-\color{blue}{\left(-\log y\right)}\right)\right) \]

      4. remove-double-neg 91.4%

         \[\leadsto y - \left(z + y \cdot \color{blue}{\log y}\right) \]

   7. Simplified 91.4%

      \[\leadsto y - \left(z + \color{blue}{y \cdot \log y}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+41}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+169}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(y - z\right) - \log y \cdot \left(y + 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (- (- y z) (* (log y) (+ y 0.5)))))

C

double code(double x, double y, double z) {
	return x + ((y - z) - (log(y) * (y + 0.5)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - z) - (log(y) * (y + 0.5d0)))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y - z) - (Math.log(y) * (y + 0.5)));
}

Python

def code(x, y, z):
	return x + ((y - z) - (math.log(y) * (y + 0.5)))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y - z) - Float64(log(y) * Float64(y + 0.5))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y - z) - (log(y) * (y + 0.5)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - z), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

   2. associate-+l- 99.9%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

4. Final simplification 99.9%

   \[\leadsto x + \left(\left(y - z\right) - \log y \cdot \left(y + 0.5\right)\right) \]

Alternative 7: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -190 \lor \neg \left(x \leq 40\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -190.0) (not (<= x 40.0))) (- x z) (- (* (log y) -0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -190.0) || !(x <= 40.0)) {
		tmp = x - z;
	} else {
		tmp = (log(y) * -0.5) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-190.0d0)) .or. (.not. (x <= 40.0d0))) then
        tmp = x - z
    else
        tmp = (log(y) * (-0.5d0)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -190.0) || !(x <= 40.0)) {
		tmp = x - z;
	} else {
		tmp = (Math.log(y) * -0.5) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -190.0) or not (x <= 40.0):
		tmp = x - z
	else:
		tmp = (math.log(y) * -0.5) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -190.0) || !(x <= 40.0))
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(log(y) * -0.5) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -190.0) || ~((x <= 40.0)))
		tmp = x - z;
	else
		tmp = (log(y) * -0.5) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -190.0], N[Not[LessEqual[x, 40.0]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -190 or 40 < x

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.9%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   4. Taylor expanded in z around inf 79.8%

      \[\leadsto x - \color{blue}{z} \]

   if -190 < x < 40

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + \left(y - z\right) \]

      3. associate-+l+ 99.8%

         \[\leadsto \color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)\right)} \]

      4. *-commutative 99.8%

         \[\leadsto x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + \left(y - z\right)\right) \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + \left(y - z\right)\right) \]

      6. fma-def 99.8%

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]

      7. +-commutative 99.8%

         \[\leadsto x + \mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y - z\right) \]

      8. distribute-neg-in 99.8%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y - z\right) \]

      9. unsub-neg 99.8%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y - z\right) \]

      10. metadata-eval 99.8%

          \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]

   4. Taylor expanded in y around 0 68.7%

      \[\leadsto \color{blue}{\left(x + -0.5 \cdot \log y\right) - z} \]

   5. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   6. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   7. Simplified 68.4%

      \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -190 \lor \neg \left(x \leq 40\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]

Alternative 8: 49.0% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+68}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.5e+23) x (if (<= x 1.15e+68) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e+23) {
		tmp = x;
	} else if (x <= 1.15e+68) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-7.5d+23)) then
        tmp = x
    else if (x <= 1.15d+68) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e+23) {
		tmp = x;
	} else if (x <= 1.15e+68) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.5e+23:
		tmp = x
	elif x <= 1.15e+68:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.5e+23)
		tmp = x;
	elseif (x <= 1.15e+68)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.5e+23)
		tmp = x;
	elseif (x <= 1.15e+68)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.5e+23], x, If[LessEqual[x, 1.15e+68], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999987e23 or 1.15e68 < x

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. sub-neg 99.9%

         \[\leadsto \color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + \left(y - z\right) \]

      3. associate-+l+ 100.0%

         \[\leadsto \color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)\right)} \]

      4. *-commutative 100.0%

         \[\leadsto x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + \left(y - z\right)\right) \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + \left(y - z\right)\right) \]

      6. fma-def 100.0%

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]

      7. +-commutative 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y - z\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y - z\right) \]

      9. unsub-neg 100.0%

         \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y - z\right) \]

      10. metadata-eval 100.0%

          \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]

   4. Taylor expanded in x around inf 68.2%

      \[\leadsto \color{blue}{x} \]

   if -7.49999999999999987e23 < x < 1.15e68

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

      2. associate-+l- 99.8%

         \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 99.2%

         \[\leadsto x - \left(\color{blue}{\left(\sqrt{y + 0.5} \cdot \sqrt{y + 0.5}\right)} \cdot \log y - \left(y - z\right)\right) \]

      2. associate-*l* 99.3%

         \[\leadsto x - \left(\color{blue}{\sqrt{y + 0.5} \cdot \left(\sqrt{y + 0.5} \cdot \log y\right)} - \left(y - z\right)\right) \]

      3. fma-neg 99.3%

         \[\leadsto x - \color{blue}{\mathsf{fma}\left(\sqrt{y + 0.5}, \sqrt{y + 0.5} \cdot \log y, -\left(y - z\right)\right)} \]

   5. Applied egg-rr 99.3%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(\sqrt{y + 0.5}, \sqrt{y + 0.5} \cdot \log y, -\left(y - z\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \color{blue}{\log y \cdot \sqrt{y + 0.5}}, -\left(y - z\right)\right) \]

      2. neg-sub0 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, \color{blue}{0 - \left(y - z\right)}\right) \]

      3. metadata-eval 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, \color{blue}{\log 1} - \left(y - z\right)\right) \]

      4. sub-neg 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, \log 1 - \color{blue}{\left(y + \left(-z\right)\right)}\right) \]

      5. +-commutative 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, \log 1 - \color{blue}{\left(\left(-z\right) + y\right)}\right) \]

      6. associate--r+ 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, \color{blue}{\left(\log 1 - \left(-z\right)\right) - y}\right) \]

      7. metadata-eval 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, \left(\color{blue}{0} - \left(-z\right)\right) - y\right) \]

      8. neg-sub0 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, \color{blue}{\left(-\left(-z\right)\right)} - y\right) \]

      9. remove-double-neg 99.3%

         \[\leadsto x - \mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, \color{blue}{z} - y\right) \]

   7. Simplified 99.3%

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(\sqrt{y + 0.5}, \log y \cdot \sqrt{y + 0.5}, z - y\right)} \]

   8. Taylor expanded in z around inf 41.5%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   9. Step-by-step derivation

      1. mul-1-neg 41.5%

         \[\leadsto \color{blue}{-z} \]

   10. Simplified 41.5%

       \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+68}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 58.3% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ x - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x z))

C

double code(double x, double y, double z) {
	return x - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - z
end function

Java

public static double code(double x, double y, double z) {
	return x - z;
}

Python

def code(x, y, z):
	return x - z

Julia

function code(x, y, z)
	return Float64(x - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - z;
end

Wolfram

code[x_, y_, z_] := N[(x - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

   2. associate-+l- 99.9%

      \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x - \left(\left(y + 0.5\right) \cdot \log y - \left(y - z\right)\right)} \]

4. Taylor expanded in z around inf 58.9%

   \[\leadsto x - \color{blue}{z} \]

5. Final simplification 58.9%

   \[\leadsto x - z \]

Alternative 10: 30.2% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)} \]

   2. sub-neg 99.8%

      \[\leadsto \color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + \left(y - z\right) \]

   3. associate-+l+ 99.9%

      \[\leadsto \color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + \left(y - z\right)\right)} \]

   4. *-commutative 99.9%

      \[\leadsto x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + \left(y - z\right)\right) \]

   5. distribute-rgt-neg-in 99.9%

      \[\leadsto x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + \left(y - z\right)\right) \]

   6. fma-def 99.9%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y - z\right)} \]

   7. +-commutative 99.9%

      \[\leadsto x + \mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y - z\right) \]

   8. distribute-neg-in 99.9%

      \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y - z\right) \]

   9. unsub-neg 99.9%

      \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y - z\right) \]

   10. metadata-eval 99.9%

       \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y - z\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]

4. Taylor expanded in x around inf 29.6%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 29.6%

   \[\leadsto x \]

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- (+ y x) z) (* (+ y 0.5) (log y))))

C

double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * log(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y + x) - z) - ((y + 0.5d0) * log(y))
end function

Java

public static double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * Math.log(y));
}

Python

def code(x, y, z):
	return ((y + x) - z) - ((y + 0.5) * math.log(y))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y + x) - z) - Float64(Float64(y + 0.5) * log(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y + x) - z) - ((y + 0.5) * log(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision] - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))